Don't bother.
You can have a doppler (ultrasound) of your carotid artery to see if there's any problem there.
Problem is, the way ultrasounds work, they just measure the speed of the blood rushing through the artery towards your brain and back.
Then the doctor compares that speed to some common values and if you're within the proper range, you're cool.
The only way to get a diagnosis through a carotid doppler is to have a previous one (say from when you didn't have tinnitus) so the doctor can compare the two and (maybe) find something which has been changing for the worst.
Unlikely.
Same thing with MRI.
It simply shows some images of areas of interest to the radiologist. Now, certain artifacts on those images like lesions or tumors show pretty well, but ear issues? No. The brain MRI is a very useless exam for tinnitus - excluding acoustic neuroma which is like the rarest thing out there and usually benign.
Maybe the best fucking radiologist in the world can make something out but even that is a slim chance.
Again, if you had an older MRI for comparison...
One last test is the MRA, which is kinda like an MRI but the machine makes images of the blood vessels running around your brain. Still extremely difficult to diagnose anything unless there is something hideously obvious like a vessel twisted around your acoustic nerve or some gigantic fistula in your cochlea.
A CT scan will not be any more useful than an MRI, besides giving a pretty hefty amount of radiation.
Sorry for the pessimistic answers but the simple truth is that current imaging techniques were not designed for ear issues in the first place and are only performed so that your doctor can rule out something extremely serious.
However, you need to talk to your doc about your options since (real) pulsatile tinnitus might actually have an underlying cause.