Use "whom" in emails?

Solution 1:

In my opinion, it depends entirely upon the intended audience.

For a business email, especially one in a 'formal' office (e.g. a government office, a Fortune 100 company, or any office that has people regularly appearing in suit jackets) it is, I think, entirely appropriate to use "whom."

For a personal email, the expectations are rather lower; it depends on your audience and your relationship with them--regardless of my ability, my SO and I have built a sort of private jargon that bears little resemblence to properly written English; "whom" does not often appear in our private correspondance.

For a business email in a less-than-formal setting (or an informal email between close business associates in a more formal setting) similar guidelines may apply.

As I work in an office more to the formal side of the continuum, I have been known to use 'whom' on occasion; nobody has complained yet.

Solution 2:

Most people don't know how to use whom and get it entirely wrong, but that doesn't mean it's archaic. Correct use of who/whom/whose is just as important as he/him/his, she/her/hers and it/it/its.

Your example, “the guy whom you said would be calling”, is not correct. The simplest way to see that is to replace who with he; your example then becomes, “you said him would be calling”, which is obviously wrong. The correct form would be “you said he would be calling”, so the original should be “who you said would be calling”.
(See also moioci's answer here.)

Having said that, in an informal email, I would be inclined to skip the whole issue and use: “I chatted with the guy you said would be calling.”