I'm getting a rattle/rumble for about 2-3 seconds on a cold start on my 2001 4.0 with 148k miles.
This only started after changing several things at one time.
It does this 90% of the time on cold starts, but doesn't seem to do it every time.

I did headgaskets (Victor Reinz), oil pump gears (BP Utah), and timing chain (Genuine).
The heads were done at a local machine shop.
I use Shell Rotella 15W40 with Purolator L40316.

I changed out the oil filter with another L40316, with no difference.
The engine is acting normal otherwise, with no abnormal noises and runs well.
I haven't checked the pressure yet.

I hate to say it, but I've put a few thousand miles on it since it's been doing this, with no change (good or bad).

It almost sounds like the lifters are not getting oil right away.
Has anyone experienced anything similar?

Any insight would be appreciated.
Thanks

 

I had a somewhat similar issue. On cold starts I would notice a slight rattle and also that my oil pressure light was slow to go out. I too suspected the lifters/rockers.

I had recently done the HGs, oil pump/timing chain, etc. For me this went away after my first oil change after that job.

 

I had good luck adding Turbo Maxx to my engine oil to quiet my rockers/lifters.

I have no affiliation with them. Just used their product.