miRNAs are indeed small non-coding RNAs. We can detect them by sequencing all RNAs in a given size range, map them to the corresponding genome, and look for the places where reads stack up in a pattern indicating that it's a miRNA precursor (basically: two stacks of reads with a given spacing, precise drop-off in read coverage that indicates precise Dicer processing, the resulting sequence folds into a stable hairpin)

That pattern may, however, also be found at sites generating other types of small structural RNAs, e.g. tRNAs. Removing all sites/sequences known to generating other types of structural RNAs is an efficient heuristic that will remove a lot of "false positives" that we would otherwise have to identify as "non-miRNAs" by close inspection of the data, or might miss altogether.

In a recent study where we detected miRNAs in a non-model organism (but with a sequenced genome), we did it as follows:

Abundant noncoding RNAs (rRNAs, tRNAs, snRNAs, and snoRNAs) were annotated to the N. vectensis genome using BLASTN (version 2.2.18) to identify homologs of

1. Drosophila melanogaster annotated rRNAs, tRNAs, snRNAs, and snoRNAs (downloaded from FlyBase and GenBank),
2. the most abundant rRNA variants found in previous studies involving sequencing of small RNAs (Ghildiyal et al. 2008; Seitz et al. 2008),
3. Homo sapiens rRNAs, snRNAs, and snoRNAs (downloaded from GenBank, and the LBME human snoRNA database), and
4. the complete set of tRNAs found in the Genomic tRNA database.

Genomic sequences with a significant BLAST hit (E-value < 10^-4) to any of those databases were flagged as "abundant noncoding RNAs," and deep-sequencing reads showing perfect identity to one of those were discarded.