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Basically, you can think of there being two parts of the immune system that are wiped out before the transplant - the stem cell part, which is the starting material to make the immune system, and the "memory" immune part, which are the cells that "remember" your past infections and vaccines. The person getting the transplant only gets the stem cells, so they don't have any of the "memory" part. Luckily, the new stem cells can re-make the whole immune system, but you need to re-teach the new stem cells (get new vaccines).

Immune memory is not stored in bone marrow stem and progenitor cells. These are the only cells that expand to reconstitute the recipient's immune system.

You don't transplant memory cells that live in secondary lymphoid organs like the spleen and lymph nodes.

New memory is trained in the thymus to teach t cells not to attack self. Old memory from the donor would attack the recipient (graft vs host disease) which is a bit of what you want to hunt down the remaining cancer cells

What the others have said is the correct answer to your question in the text.

However I'd just like to add that in the case of an allogeneic hematopoietic stem cell transplant (with a donor) you actually do "inherit the immune system of the donor" - sorta...

When you receive stem cells, those stem cells originate from another body, which the cells are completely compatible with. Should you later down the road need a new kidney for example, the original donor can donate their kidney without the need of immunosuppresants. Normally you'd need immunosuppresants for as long as the kidney was in your body in order to avoid rejection. However since the immune systems cells and the kidney are compatible - you don't.

I suppose it's exceedingly rare though. Just something I remember from immunology classes that I found extremely interesting.

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Since the lymphoid organs stay (and are the memory cells), why could they not just convey the immune info to the new marrow and progenitor cells?

If you can and dont mind, would you explain this a little simplified?

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One of the big reasons why allogenic bone marrow transplant works. The recipient gets chemo and radiation to kill off all of their own stem cells, which are mostly cancerous by this time. Not all die off and some may survive to cause a relapse of the leukemia.

Even in a 10/10 mhc match, the donor cells can't be perfectly matched to everything, so some graft vs host action is thought to be the Final element in battling whatever cancer cells may remain. To check for residual leukemia cells, they often look at the bone marrow for chimera - % of graft vs pretransplant stem cells....

Hope that answers your question