If the files are in mp3 format then you should be able to make a direct copy or even copy them to your computer hard drive, and then burn them to a new blank CD - with any burning package.
Some players are picky about the type of CD R media they will play. I'd recommend trying some good quality media like Verbatim etc.
Also - (probably most importantly) not all stand alone CD player will play mp3 files. Make sure you burn them as CDAudio format - usually this will happen by default if you select the "burn audio CD" option in your burning application (but not always)