Like Backwoodsman, my experience is limited to the UK. However it is my belief that you have to get the licence in the country where you are publishing the CD. There is some information about mechanical licensing in the USA here:

Obtaining copyright permissions

"Mechanical licenses apply to reproduction and distribution of musical works on records and CDs, in permanent digital downloads, and in some other digital uses. For example, these licenses permit subsequent recording artists to “cover” songs written by others. The royalty rates for records, CDs, and permanent digital downloads are set by statute, and the licenses are compulsory. The Harry Fox Agency is the primary source for mechanical licenses in the United States"

I suggest you contact Harry Fox Agency for advice. Harry Fox uses a system called Songfile for mechanical licenses:
Songfile

If I am correct it should not be necessary to contact a UK agency (which is MCPS for mechanical licenses) or the publisher unless Harry Fox cannot grant it.

You should also be aware that you own the copyright of your own performance on the CD - this is quite separate from any composers' copyright. If your CD is played in public or broadcast you are entitled to be paid. The royalty is shared between the publisher and the performers, so if you are self-releasing then you may have to register as both. In the UK the relevant organisation is PPL, in the US I think it is Sound Exchange. I wish someone had told me about this when I first released an album, as one track got played regularly on the BBC, probably because one of the composers was the programme's presenter. He would have received royalties for this, we didn't get a penny because we didn't know about PPL and hadn't registered with it.