The Shulhan Aruch (Orah Haim 208:7) writes that blessing over rice is “HaAdama”. However, if rice is ground into flour and is baked, the blessing would be “Mezonot” since such flour has a satiating quality similar to other grains but the after-blessing would remain “Bore Nefashot”, nonetheless. The Rama (ibid.) comments that as longs as the rice becomes mushy and soft, the blessing would be “Mezonot”. Other Poskim, following the Shulhan Aruch’s lack of such a stipulation, say that the blessing of rice is always “Mezonot”, as long as it is cooked. Rabbi Ovadia Yosef (Hazon Ovadia, pg. 183) and many other Sephardic Poskim write that as long as rice is cooked, it would have the blessing of “Mezonot”. Furthermore, Rabbi Itzhak Krispin (Shemo Itzhak, pg. 4) confirms that this is also the custom of the Moroccan community.
An exception to this rule is if the chaff is still on the rice, in which case the blessing would be “HaAdama”. Brown rice is mistakenly thought to contain the chaff and therefore the blessing of “HaAdama”. Nevertheless, Rabbi Alexander Mandelbaum (Vezot HaBeracha, Birure Halacha § 17) proves that conventional brown rice does not contain the chaff and simply contains a brown layer which covers the white part of the rice grain. Therefore, even cooked brown rice has the blessing of “Mezonot”.
Regarding rice cakes, the rice grains are not cooked but rather are puffed through heat. Some Poskim say that since the rice was converted into an edible form, regardless of the process, the blessing would be “Mezonot”. Nevertheless, Rabbi Ovadia Yosef (Hazon Ovadia, pg. 184), Rabbi Ben Zion Abba Shaul and Rabbi Shlomo Zalman Auerbach all rule that, since the rice is not actually cooked but merely puffed, the blessing would be “HaAdama”.
Summary: The blessing for both white and brown rice, whether cooked or ground into flour, is “Mezonot”. The blessing over rice cakes is “HaAdama”.