At level 3, new summoners get a 400 RP bonus. Are there other ways to earn RP using such bonuses or promotions like the Tristana and Alistar free promos?
The only RP you'll get for free is the 400 RP upon hitting level 3. Plus the 450 RP Riot usually hands out in December as a Christmas gift. The new recruitment program also rewards heaps of RP and IP.
The refer-a-friend program can get you 975 RP (and 5000 IP). Riot has cracked down on people making 10 alts and grinding them up to level 10 just for the free stuff, but it's a bit of a no-brainer if you happen to know 10 or more people who don't (yet) play LoL :)
For more info:
As others have said there isn't a true way to get free Riot Points other then the refer-a-friend bonus.
I have gotten some free Riot Points from a few giveaways people have hosted (mostly YouTube/stream related to grow their audience).
I recommend following this website's facebook page: They regularly post such giveaways for you to enter, the rest is luck though really.
Alternatively you can join a rewards site like PrizeRebel or GamerRewards and complete surveys for the RP but that takes a long time.
Hope that helps!