Just to add to the answers already given:

• One of the silencing mechanisms for miRNA's can result in breakdown of mRNA, which will of course affect the amount of mRNA measured (the mRNA measurement in that case detects the net effect of transcription and miRNA regulation).
• I recently heard a brilliant talk by Mats Lundman (opening talk at SBMCTA in Sao Pedro, Brazil) who showed a mechanism where miRNA are transcribed downstream of the mRNA they are targeted against, thereby leading to a kind of burst time in mRNA availability. So I don't fully agree with @lyco's remark that different regulations mostly are in the same direction. Fine tuning may actually be opposite and time regulated.

No, I don't think that this is correct. (Disclosure: I do microarrays for a living so you might consider me biased). Without any doubt is there posttranscriptional regulations by miRNAs, but in most cases this contribution is minor. There are other posttranslational factors that govern protein levels, which are independent from the mRNA amount and thus cannot be measured by transcriptomics approaches (most important: protein half life, but also various sources of translational regulation.

• Thus, if you are interested in protein levels, microarrays are not the way to go.
• If you are interested in 'protein level changes', microarrays are not optimal but can give a reasonable approximation. Note: if there is a biologically meaningful induction of a gene's mRNA, this is rarely counteracted by posttranscriptional factors - often transcriptional and translational controls point in the same direction.
• Finally, there are a number of questions where microarray are very good: responses to drugs and toxins (often regulated on the mrNA level), assessment and specificity of epigenetic silencing, mRNA stability (n.b: microarrays measure steady state mRNA levels, not transcriptional rate) and general 'transcriptional responses' (to inflammation, DNA damage, senescences, ...)

Compare this related blog post

1. I would not say that microarrays "often only show vague changes in gene expression." Depending on how the experiment is designed, very significant fold-changes can be observed. What is true is that often, only a relatively small fraction of probes show differential expression, but that's exactly what you'd expect if a subset of genes in particular pathways are responding to changed conditions.
2. Microarrays do not "come with" miRNA probes, but there are custom arrays and vendor platforms designed to detect miRNA. Here's an article that compares 6 such platforms.
3. Our knowledge of miRNAs is very incomplete. Many have been predicted from sequence analysis, but very few validated or assigned biological functions. There's a rather overwhelming literature on this topic, but I'd start with some miRNA databases such as microRNA.org.
4. I think 3'-UTR binding would give a very noisy signal on microarrays, but see point (2), there are miRNA array platforms.

Actually 84% of miRNA silencing is by mRNA degradation, which would be picked up by microarrays. See this paper: http://www.nature.com/nature/journal/v466/n7308/full/nature09267.html