Are there any libraries, pieces of code or suchlike that'll let me play ZX Spectrum .ay files in my XNA Game Studio games?
If you need sound in XNA, and want XBOX 360 support you need to use the supported file formats. You are properbly better of by trying to convert the .ay files to a format that XNA natively support.
If you only want to support Windows, then search for a .NET Library that can play them, it will work in XNA on windows if it works in .NET.