Others will be more knowledgeable on this topic, but IMO you don't want your engine shaking -- not good for the things down stream: gear box, shaft, stuffing box/seals, cutlass bearing, prop etc. Have a mechanic check it out. You're guess re the mounts may be right on.
But diesels do clatter and shake a bit more than a gasoline engine will. Can you give a little more detail? Just how violet is the shaking? Can you estimate the distance in inches that say, the oil cap, is moving within at the worste idle?
My MD11C volvo shakes a lot. As you said certain RPMs are worse than others at about 700 mine is probably worst. Above 1200 it smooths out a lot. 2 cylinders shake more than three!!! My brother had a single cyclinder diesel and he said that it really shook..a small yanmar.
My Volvo has some rough spots as well which smooth out at higher and lower revs. Pretty sure that it's just an idiosyncracy. The engine and drive train get inspected by a good diesel mechanic every spring and fall.
Pub may be on to something... if it only occurs when in gear. Failing that I'd take a real good look at your mounts.. an inch and a half is a bit much, and if the shaft is visibly moving about (rather than just turning)that's hard on other things like struts and shafts and shaft seals.
Although most folks never replace motor mounts, the foks who make them say they should be replaced every 5 years for conventional rubber mounts, longer for some synthetic materials, just like every other rubber part on the engine.
I'd suggest checking the shaft and prop to make sure there is no ball of barnacles throwing it off balance, and then eyeball the motor mounts to see if there is anything cracked, or shifted (leaving a mark where it moved). All things considered, it might pay to have someone take a look at it hands-on if you don't find anything on the prop.
I've suffered with engine ills and mechanic diagnosis for some time. So I can remotely diagnose this very easily.
Since this seems like a off-balance issue and you will spend 6 months taking it to 12 mechanics with 13 different answers. I will save you some time. It has to be either:
Bad bulb in the oven.
Bad injector that works better at higher flow rates.
Back pressure from the exhaust which when fixed no one will be able to tell you how it had anything to do with it.
In other words there is no way it could be:
fishing line on the shaft
Because all those things would make sense.
So save yourself some time and don't even bother with what makes sense at first.
You will be lucky if after it is fixed you even have a guess as to why it worked.
If it is only jumping around and inch and a half you could just wait for it to get to two and half inches. Then it will be out of the boat where it is easier to service.
If your engine is a single cylinder, it will generally vibrate more at lower RPMs as it is basically running unbalanced in some ways... the more cylinders an engine has, generally the smoother it operates. You probably should replace the engine mounts.
I'm not very familiar with that model, but what is the idle RPM supposed to be? You said that it smoothes out at 1000, if I remember correctly isn't idle speed about 900 on that model? My Volvo MD31A (62hp) shakes and vibrates when it first starts and is below normal idle speed and then smoothes out. You should contact Volvo and get the idle setting, then check the RPM at the engine with a handheld tach, don't believe your tachometer until you verify that it's correct. It may just be that your idle is set too low. I had a Yanmar three cylinder that would shake pretty violently at low RPM when cold, but was fine after running for a minute or so.
If you have a small air leak in the fuel system often a diesel will idle poorly and run poorly at low rpm (misfire and shake). Once you throttle up it can greatly improve the performance. I'd guess a small air leak is by far the most likely cause of the shaking.
Your idle speed may be too low and Your tachometer may be off. If you can find a laser revolution counter, It will tell you your true engine speed and then compare it to the engine tach. It may be off. And you may be to slow causing more vibration.
I am not a Volvo mechanic, but I will relate to you what happen to me with my Yanmar. About the 3th year we owned the boat the Yanmar started shaking so bad at idle the whole rig would shake. It sounded terrible. What was wrong was the throttle cable has stretched and it was idling to slow, in fact it was firring only one cylinder. I made a adjustment on the cable, no more problem.