You should type:

heatmap

copy the function to a new script "heatmap2" like so:

heatmap2 <- function(...... ## from heatmap

add this line right before the final brace:

return(rev(labRow))

then call your same heatmap function but assign it to a variable:

t <- heatmap2(dm[pvals<0.06,])

This will output the row names used for the heatmap into variable t in your local environment. You can repeat for column names if you'd like, or add the variable assignment explicitly to the function.

I'm not sure what you mean about row names by cluster.

Here is the full code for the heatmap2 function below. I'm not sure if the copy/paste will work from Biostar's site. It will be super messy in any case, so I recommend doing it yourself as detailed above:

heatmap2 <- function (x, Rowv = NULL, Colv = if (symm) "Rowv" else NULL,
                      distfun = dist, hclustfun = hclust, reorderfun = function(d,
                                                                                w) reorder(d, w), add.expr, symm = FALSE, revC = identical(Colv, 
                                                                                                                                           "Rowv"), scale = c("row", "column", "none"), na.rm = TRUE, 
                      margins = c(5, 5), ColSideColors, RowSideColors, cexRow = 0.2 + 
                              1/log10(nr), cexCol = 0.2 + 1/log10(nc), labRow = NULL, 
                      labCol = NULL, main = NULL, xlab = NULL, ylab = NULL, keep.dendro = FALSE, 
                      verbose = getOption("verbose"), ...) 
{
        scale <- if (symm && missing(scale)) 
                "none"
        else match.arg(scale)
        if (length(di <- dim(x)) != 2 || !is.numeric(x)) 
                stop("'x' must be a numeric matrix")
        nr <- di[1L]
        nc <- di[2L]
        if (nr <= 1 || nc <= 1) 
                stop("'x' must have at least 2 rows and 2 columns")
        if (!is.numeric(margins) || length(margins) != 2L) 
                stop("'margins' must be a numeric vector of length 2")
        doRdend <- !identical(Rowv, NA)
        doCdend <- !identical(Colv, NA)
        if (!doRdend && identical(Colv, "Rowv")) 
                doCdend <- FALSE
        if (is.null(Rowv)) 
                Rowv <- rowMeans(x, na.rm = na.rm)
        if (is.null(Colv)) 
                Colv <- colMeans(x, na.rm = na.rm)
        if (doRdend) {
                if (inherits(Rowv, "dendrogram")) 
                        ddr <- Rowv
                else {
                        hcr <- hclustfun(distfun(x))
                        ddr <- as.dendrogram(hcr)
                        if (!is.logical(Rowv) || Rowv) 
                                ddr <- reorderfun(ddr, Rowv)
                }
                if (nr != length(rowInd <- order.dendrogram(ddr))) 
                        stop("row dendrogram ordering gave index of wrong length")
        }
        else rowInd <- 1L:nr
        if (doCdend) {
                if (inherits(Colv, "dendrogram")) 
                        ddc <- Colv
                else if (identical(Colv, "Rowv")) {
                        if (nr != nc) 
                                stop("Colv = \"Rowv\" but nrow(x) != ncol(x)")
                        ddc <- ddr
                }
                else {
                        hcc <- hclustfun(distfun(if (symm) 
                                x
                                else t(x)))
                        ddc <- as.dendrogram(hcc)
                        if (!is.logical(Colv) || Colv) 
                                ddc <- reorderfun(ddc, Colv)
                }
                if (nc != length(colInd <- order.dendrogram(ddc))) 
                        stop("column dendrogram ordering gave index of wrong length")
        }
        else colInd <- 1L:nc
        x <- x[rowInd, colInd]
        labRow <- if (is.null(labRow)) 
                if (is.null(rownames(x))) 
                        (1L:nr)[rowInd]
        else rownames(x)
        else labRow[rowInd]
        labCol <- if (is.null(labCol)) 
                if (is.null(colnames(x))) 
                        (1L:nc)[colInd]
        else colnames(x)
        else labCol[colInd]
        if (scale == "row") {
                x <- sweep(x, 1L, rowMeans(x, na.rm = na.rm), check.margin = FALSE)
                sx <- apply(x, 1L, sd, na.rm = na.rm)
                x <- sweep(x, 1L, sx, "/", check.margin = FALSE)
        }
        else if (scale == "column") {
                x <- sweep(x, 2L, colMeans(x, na.rm = na.rm), check.margin = FALSE)
                sx <- apply(x, 2L, sd, na.rm = na.rm)
                x <- sweep(x, 2L, sx, "/", check.margin = FALSE)
        }
        lmat <- rbind(c(NA, 3), 2:1)
        lwid <- c(if (doRdend) 1 else 0.05, 4)
        lhei <- c((if (doCdend) 1 else 0.05) + if (!is.null(main)) 0.2 else 0, 
                  4)
        if (!missing(ColSideColors)) {
                if (!is.character(ColSideColors) || length(ColSideColors) != 
                            nc) 
                        stop("'ColSideColors' must be a character vector of length ncol(x)")
                lmat <- rbind(lmat[1, ] + 1, c(NA, 1), lmat[2, ] + 1)
                lhei <- c(lhei[1L], 0.2, lhei[2L])
        }
        if (!missing(RowSideColors)) {
                if (!is.character(RowSideColors) || length(RowSideColors) != 
                            nr) 
                        stop("'RowSideColors' must be a character vector of length nrow(x)")
                lmat <- cbind(lmat[, 1] + 1, c(rep(NA, nrow(lmat) - 1), 
                                               1), lmat[, 2] + 1)
                lwid <- c(lwid[1L], 0.2, lwid[2L])
        }
        lmat[is.na(lmat)] <- 0
        if (verbose) {
                cat("layout: widths = ", lwid, ", heights = ", lhei, 
                    "; lmat=\n")
                print(lmat)
        }
        dev.hold()
        on.exit(dev.flush())
        op <- par(no.readonly = TRUE)
        on.exit(par(op), add = TRUE)
        layout(lmat, widths = lwid, heights = lhei, respect = TRUE)
        if (!missing(RowSideColors)) {
                par(mar = c(margins[1L], 0, 0, 0.5))
                image(rbind(if (revC) 
                        nr:1L
                        else 1L:nr), col = RowSideColors[rowInd], axes = FALSE)
        }
        if (!missing(ColSideColors)) {
                par(mar = c(0.5, 0, 0, margins[2L]))
                image(cbind(1L:nc), col = ColSideColors[colInd], axes = FALSE)
        }
        par(mar = c(margins[1L], 0, 0, margins[2L]))
        if (!symm || scale != "none") 
                x <- t(x)
        if (revC) {
                iy <- nr:1
                if (doRdend) 
                        ddr <- rev(ddr)
                x <- x[, iy]
        }
        else iy <- 1L:nr
        image(1L:nc, 1L:nr, x, xlim = 0.5 + c(0, nc), ylim = 0.5 + 
                      c(0, nr), axes = FALSE, xlab = "", ylab = "", ...)
        axis(1, 1L:nc, labels = labCol, las = 2, line = -0.5, tick = 0, 
             cex.axis = cexCol)
        if (!is.null(xlab)) 
                mtext(xlab, side = 1, line = margins[1L] - 1.25)
        axis(4, iy, labels = labRow, las = 2, line = -0.5, tick = 0, 
             cex.axis = cexRow)
        if (!is.null(ylab)) 
                mtext(ylab, side = 4, line = margins[2L] - 1.25)
        if (!missing(add.expr)) 
                eval(substitute(add.expr))
        par(mar = c(margins[1L], 0, 0, 0))
        if (doRdend) 
                plot(ddr, horiz = TRUE, axes = FALSE, yaxs = "i", leaflab = "none")
        else frame()
        par(mar = c(0, 0, if (!is.null(main)) 1 else 0, margins[2L]))
        if (doCdend) 
                plot(ddc, axes = FALSE, xaxs = "i", leaflab = "none")
        else if (!is.null(main)) 
                frame()
        if (!is.null(main)) {
                par(xpd = NA)
                title(main, cex.main = 1.5 * op[["cex.main"]])
        }
        invisible(list(rowInd = rowInd, colInd = colInd, Rowv = if (keep.dendro && 
                                                                            doRdend) ddr, Colv = if (keep.dendro && doCdend) ddc))
        return(rev(labRow))
}

I have run into issues with extracting clusters directly from the heat map function. Thus, I prefer to cluster using hclust(), and then pass in the hclust object as a dendrogram to manually to the plotting function to set the order of the rows/cols. You can then use cutree() on the hclust object to get any number of clusters you want. Just adjust the dist() parameters to calculate dissimilarity differently, and adjust the hclust() parameters to cluster differently.

hc <- hclust(dist(my_matrix, method = "euclidean"), method = "complete")
ct <- cutree(hc, k = num_clusters)
heatmap.2(my_matrix, Colv = as.dendrogram(hc), trace="none")

The cluster core of heatmap is two parameters hclustfun and distfun (see ?heatmap). So just use the same clustering methods and distfun distance independently, you will get the same clustering result and then get the genes in clusters.