Untrue.
The RX-7's rotary engine (the 13B series) is actually phenomenally reliable, and there are many examples with well over 300k road miles. The RX-8 uses a newer variant of the same engine - the 13B-MSP RENESIS. Additionally, Mazda used the rotary in competition (in the infamous Mazda 787B) at the 24 hours of LeMans, which is a notoriously difficult test of engine reliability.
The myth of poor rotary reliability primarily stems from the 3rd-generation RX-7. The notoriously-complex mechanically-controlled sequential twin-turbo system (affectionately known as "the rat nest") was prone to hose disconnects and failures, which often led to over-boost situations. Additionally, owners that modify their cars often exacerbated the problem, installing free-flowing intakes and exhausts, as well as boost controllers, and did not properly modify air-fuel ratios to compensate. These situations led to increased engine heat and premature failure, which gave the rotary an unfair bad reputation. Many owners now convert to a simpler single-turbo setup with electronic boost control and careful air-fuel tuning, which is far more reliable.
The actual (and admittedly significant) drawbacks of rotary engines are poor fuel efficiency and high exhaust emissions, both due to incomplete combustion. This has kept them out of mainstream applications and thus they tend to be used primarily in performance applications.
