The correct way to distribute this app would have been to provide a PPA you could put in your apt sources, which would be updated for each new version of Ubuntu, and allow the full package management capability. By using .debs any dependencies need to be manually satisfied, and you can't push updates or bug-fixes, although it looks like Amazon has written that functionality into the program itself. The deb also depends on very specific versions of libboost, rather than just specifying a minimum version number, which would allow it to work on later releases.
So installing it on lucid, or any other version is a bit of a pain. Here is one solution, that grabs the original dependencies from launchpadlibrarian and installs them:
wget https://launchpadlibrarian.net/26959932/libboost-signals1.34.1_1.34.1-16ubuntu1_i386.deb \ https://launchpadlibrarian.net/26959936/libboost-thread1.34.1_1.34.1-16ubuntu1_i386.deb \ https://launchpadlibrarian.net/26959922/libboost-iostreams1.34.1_1.34.1-16ubuntu1_i386.deb \ https://launchpadlibrarian.net/26959918/libboost-filesystem1.34.1_1.34.1-16ubuntu1_i386.deb \ https://launchpadlibrarian.net/26959916/libboost-date-time1.34.1_1.34.1-16ubuntu1_i386.deb \ https://launchpadlibrarian.net/26959928/libboost-regex1.34.1_1.34.1-16ubuntu1_i386.deb \ https://launchpadlibrarian.net/34165098/libicu40_4.0.1-2ubuntu2_i386.deb sudo dpkg -i *.deb sudo dpkg -i amazonmp3.deb