Search one or more of the various music social networks catering specifically to musicians and music producers. Myspace Music is one of the most popular and has become a de facto music social network with the rise of Facebook. Other music social networks include Sonific and Reverb Nation. Once you find an artist you like, send them a message and tell them you're interested in using their music.
Place a music listing on Sonicbids. Sonicbids allows bands, artists, and composers to bid on gigs listed by those in the market for music. Sign up as a promoter or agency, compose your music listing, then review the electronic press kits of different artists who submit to your listing. This prevents you from having to sift through the over 300,000 bands and artists listed there.
Hire a music contractor through Elance which is a network of freelance entrepreneurs that specialize in various fields of expertise. Since members generally have to pay to bid on job listings, placing an ad is free for employers. Since there's no specific category for music under Design & Multimedia, you'll have to make sure that you list it under corresponding project that you are working on. Music contractors will bid on your project and you can listen to each to see if they meet your criteria.
Advertise on Freelancer. Freelancer is similar to Elance but with some marked differences, the first being that it's free for everyone to bid on and list projects. The site only makes money when a contractor is hired. The site then takes a certain percentage out of what the contractor gets paid. The has a Freemarket section of where you may browse certain categories and buy the rights to use music for very low fees.The site gives you the option of searching different music genres and you can click and listen to each clip.
Join music forums such as Taxi, Gearslutz, or Electronic Music. Start a thread saying your searching for instrumentals. This is not the most convenient method as you have to do a lot more leg work to make sure you're getting quality material.