Using a Micrel MIC2940A-3.3 LDO regulator without additional circuitry?
Your circuit draws 85mA maximum, so the maximum power dissipation is 490mW. That is acceptable for a TO-220 package with no heatsink. 

However, you could save your users quite a bit of money by using something like 3-4 AA cells and an LDO. There is also more choice of LDOs if the input voltage is lower, and some have low enough quiescent current that you could avoid a hard on/off switch. Or you could use a switching regulator, but the cost per kWh of 9V batteries is higher than that from AA cells, so it would be even better to use a SMPS from AA cells. 

Suggest you calculate the cost per hour of operation each way. 

By the way, the capacitors are not just to keep the MCU happy- this LDO, like most, will oscillate if you don't provide the appropriate capacitance using the appropriate type of capacitors on the output. Not paying attention to this stuff is a shortcut to a lot of headaches: 


  External Capacitors
  A 10μF (or greater) capacitor is required between the
  MIC2940A output and ground to prevent oscillations due
  to instability. Most types of tantalum or aluminum
  electrolytics will be adequate; film types will work, but
  are costly and therefore not recommended. Many
  aluminum electrolytics have electrolytes that freeze at
  about –30°C, so solid tantalums are recommended for
  operation below –25°C. The important parameters of the
  capacitor are an effective series resistance of about 5Ω
  or less and a resonant frequency above 500kHz. The
  value of this capacitor may be increased without limit.
  At lower values of output current, less output
  capacitance is required for output stability. The capacitor
  can be reduced to 3.3μF for current below 100mA or
  2.2μF for currents below 10mA. Adjusting the MIC2941A
  to voltages below 5V runs the error amplifier at lower
  gains so that more output capacitance is needed. For
  the worst-case situation of a 1.25A load at 1.23V output
  (Output shorted to a 22μF (or greater) capacitor
  should be used.


They also recommend at least a 0.22uF capacitor at the input and you only have 100nF. I suggest a 100uF/10V electrolytic in parallel with at least 100nF. Batteries greatly increase in internal resistance as they deplete.