What is the accepted way to write units when specifying multiple dimensions?

The first one is the only one that makes sense...

2. Implies cubic units cubed

3. Is needless

So obviously 1 is the clearest.

I 'll start with the one that you should DEFINITELY NOT use: $mm^3$ or $mm3$.

Probably the most widely used is the 1st. It's compact and economical and these are two of the most deciding factors in Engineering thought and practice.

To take the point one step further, if it's in an engineering drawing you don't even need to put units in. Unless, its explicitly stated - for mechanical engineering - in countries following the metric system, a dimension number always implies mm.

So that makes the third option, explicit but after a fashion "bloated" and redundant.

Cubic millimeters (mm³) would be used when describing volume of holding capacity.

In your situation your third option is correct, but use spaces: 1 mm x 2 mm x 3 mm, or 1 mm by 2 mm by 3 mm. Each number needs to have the unit follow it because 10 mm x 2 mm x 4 mm could also be written as 1 cm x 2 mm x 4 mm.

1x2x3 mm is usual. You might specify individual units if they used different multiples. For example, if you had a large sheet of thin material you could describe it as 1 m x 2 m x 3 mm. However in metric engineering drawings it is common to keep everything in mm and describe this as 1000 x 2000 x 3 (with a note in the corner of the drawing stating all dimensions in mm).

Your third format can be used for mixed units. Here in the UK I can buy wood sheet material in e.g. 8 ft x 4 ft x 18 mm sizes.

In an earlier comment, I suggested that an answer to "what is the correct way..." (as opposed to "What is the generally accepted way...") might be found in the ISO 80000-3 standard. I had a look, and there's nothing relevant in ISO 80000-3. Nor can I find any other ISO standard that explicitly states a correct way of doing this. However, ISO, when they themselves need to specify the dimensions of an object, frequently use the format in OP's third bullet point, e.g. $1,mathsf{mm}times 2,mathsf{mm}times 3,mathsf{mm}$.