Let's say I have a micro-4/3rd camera and a full frame camera, both set to 1/60 at f/2.8, taking a picture of the same scene in the same lighting. Will the exposure be the same across both cameras despite the different sensor sizes?

Yes - if it's the same lens or both lenses have the same transmission, and assuming that by saying "same exposure" you're using the same ISO rating (to even out differences in sensor efficiency).

Caveats:

• Same ISO doesn't mean same noise level.

  Different sensors operating at the same ISO level will capture different amounts of light but turn them into the same exposure. However, even though the exposure is the same, the ability to resolve detail amongst the noise will be different. The ISO rating system is designed to factor out differences in sensor efficiency so you can set any sensor regardless of size or efficiency to ISO200 and get the same exposure. To achieve this, a full frame sensor working at ISO200 is gathering a lot more light than a 4/3 sensor at ISO200 for the same scene, and it is just internally applying a different amount of gain in order to translate the scene into the same brightness values.

  All will look equivalent in the end result in terms of exposure, except that the full frame will have lower noise levels since it started with more light information. Note that there can be differences in efficiency between sensors of the same size, too; hence it's not related solely to sensor size, though that is the major factor. In short, ISO 800 in FF is the same exposure as ISO 800 in 4/3, but you'll get different noise and dynamic range on them since it's not the same sensor efficiency.

• Same f-stop doesn't necessarily mean same lens transmission.

  The common method of determining how much light comes through the lens is an f-stop. However, this measure is based on the diameter of the aperture, but does not take into account the transmissive properties of the lens elements (that is, how much light is absorbed by the glass in the lens). All lens glass absorbs some light. Modern lenses with multiple coatings absorb a good deal less, and it's not uncommon for a simple modern lens to transmit more than 99% of light.

  Without filters, the effect of transmission loss in a modern multi-coated lens is so small that in almost all cases it can be ignored, making this little more than an academic exercise with little practical value. Those cases in which it can't be ignored may include shooting for the cinema, where multiple consecutive shots should have the same exposure even though they may use a very different lens. That's why t-stops were invented; they're like f-stops by they take into account transmission properties of all your glass.