I would like to set up a project with an ATtiny chip, such as the ATtiny84 or ATtiny85, in which I interface the microcontroller directly with an XBee for both wireless serial communication, as well as monitoring a digital signal from the Xbee (digital input to the ATtiny). As the Xbee chips can handle only 3.3V and will be damaged at higher voltages (such as 5V), I was wondering whether I can simply power the entire project at 3.3V so that all digital inputs and outputs will be at that voltage?
Yes. As given in the datasheets, normal ATtinyX4/X5 chips can run with a supply between 2.7 and 5.5V inclusive, whereas the low-power versions can run between 1.8V and 5.5V. Mind the speed restrictions when running at lower voltages though; see the "Electrical Characteristics" section of each datasheet to determine the maximum characterized speed for a given voltage.