In the UK, this issue is handled by the Mechanical Copyright Protection Society (MCPS) which is in alliance with the Performing Rights Society (PRS) which licences live performances. You don't have to approach the copyright holders yourself, you just fill in a form listing all the tracks with details of the composers and arrangers, and how many CDs you'll be manufacturing. MCPS calculates the royalties due, which you pay to them and they pay the copyright owner. MCPS gives you a certificate to show the manufacturer (even if everything is non-copyright and no royalties are payable, you still go through MCPS to get the certificate). For small runs you can pay a flat fee, but if only a few of the tracks are copyright then it may be cheaper to fill in the form and calculate the actual figure.

I appreciate it is more complicated in the US as there are a number of rights agencies, but perhaps it would be simpler to do it through one of them.