Are there any rest areas in Dubai's Terminal 3?

First of all, 3 hours is trivial to spend in Terminal 3; this is because you have a significant walk from the airbridge to the transit area; and depending on when you land, you may have a long wait at the security screening area.

If you are arriving at gate A, and departing on gates B,C,D or F - you'll have to take the train to the appropriate transit lounge area - and that can easily soak up 15 - 20 minutes (if there is a crowd).

Having said that, there are rest areas (permanently reclined seating), plenty of lounges that you may access, and even a dedicated quiet area - however, from my personal experience 3 hours goes by so fast that by the time you reach the rest area, you'll find that you have to start your journey towards your gate, which may be a train ride away.

The way the terminal is designed think of it like a three lane hall.

The middle lane is the shopping area, left and right lanes are for the gates, and at the end of the three lane corridor you have additional gates.

Then there is the main concourse area (where is the large waterfall and the giant glass elevators) this is where everyone enters the transit area.

The rest areas you'll find at the gates (either on the left, right or at the end of the corridor), where you'll get (permanently fixed, but reclined) seating. For further relaxation - there is a permanent quiet area (look for signs), and of course there is the airside hotel, spa(s) (yes, more than one) and even massage parlors and of course various transit lounges.

However, again, I caution against planning for a quick nap - 3 hours is nothing for a T3 transfer.

The generic answer to "Are there any rest areas where I could [...] lie down?" is always: check Sleeping in airports. It will list the places where you can sleep -- not many at this airport -- and things to do.

Terminal 3 is dedicated for use by your carrier, Emirates Airline, and has a number of lounges available to its passengers. Serving Emirates and Qantas, and said to be the largest terminal in the world, it has dozens of services and amenities, including rest areas, shops, restaurants, and spas (three of them!). Before your arrival in Dubai, you may want to consult the terminal map so that you can navigate your way more easily during your brief, 3-hour layover.