Now you can search your favorite Hindi songs and listen to them legally on Google India Music Search. The service is providing full search for Bollywood (Hindi) old and new songs, And providing some songs of other languages like Telugu, Tamil in volumes. You can search Music based on artist, albums, movie or songs.
You can even restrict your results to a particular year, like old songs or new songs etc. The good thing about the service is, you can play the songs in single click from the search results. A small pop-up opens to play the song which you clicked. Unlike in U.S. here you can listen full song, In U.S. Google plays only preview.
It seems Google partnered with In.com, Saavn and Saregama to provide search links to the songs. if you click on the search result, a pop up will be opened from one of these sites. Google may partner with more services to provide better search for you.
For example, for search results of the latest sensational Indian film Robot, you will get the results as shown in the image below. Search results contains Robot songs in Hindi and Telugu but missed Tamil language.
You can play only one song at a time, It will be good if there is an option for playing all songs from the results, at least for albums or option for selecting multiple songs.