First of all, からして is a different grammatical construction. As Zach has already covered this in more detail in his answer, I will skip over it. And as Dave M G stated in his answer, all of these constructions have the meaning of "judging from the position of X."
Where they differ is in their usage, rather than their meaning.
～から言うと / ～から見ると
With these two, what follows must be something which contains the speaker's own thoughts, feelings, or opinion. What differentiates them is that with ～から見ると, the noun that goes in ～ may be a person, while it with ～から言うと, it may not. Examples:
- × 彼からいうと、それは間違っているそうだ。
- ○ 彼の考え方からいうと、それは間違っているそうだ。
- ○ 彼から見ると、それは間違っているそうだ。
Also seen as: ～からいえば, ～からいって, ～からいったら and ～から見れば, ～から見て, ～から見ても, ～から見たら
What differentiates ～からすると from the others is that what follows it doesn't necessarily have to be the speaker's thoughts, feelings, or opinion (but if it is, that's okay, too). Examples:
- ○ 私の予想からすると、真夏日も後数日だろう。
- × 私の予想から見ると、真夏日も後数日だろう。(× because 予想 and 見る have the same meaning here.)
- ○ いつもよく食べる彼女が食欲ないことからすると、何かあったのだろうか。
- ○ いつもよく食べる彼女が食欲ないことからすると、何かあったと思う。
Also seen as: ～からすれば, ～からしたら, ～からして
So essentially, every instance of ～からいうと can be replaced just fine with either ～からすると or ～から見ると, but the reverse is not necessarily true.
Note: According to the textbooks I used as reference when researching this answer, ～からして, while it can be used in this sense, is slightly different (as can be the other forms of ～からすると). It acts as an intensifier, basically meaning only thinking about it from that point of view.
Applying the Principles
Now, if we apply these principles to the examples given in your question, we can easily come to a conclusion on their acceptability.
- ○ 症状からすると、心臓の病気かもしれません。
- ○ 症状から見ると、心臓の病気かもしれません。
- ○ 症状から言うと、心臓の病気かもしれません。
I believe all of these are just fine, because 1) the noun prior to the expressions is not a person, and 2) what follows is the speaker's opinion. The next set is a little bit different, however.
- × 昔の人からすると、まるで別世界です。
- ○ 昔の人から見ると、まるで別世界です。
- × 昔の人から言うと、まるで別世界です。
In this case, only ～から見ると is valid, because only this expression can be from the "point of view" of a person (in this case, 昔の人).
In sum, while the meaning of these three expressions is for all intents and purposes the same, their usage is slightly different. In my opinion (自分の意見からいうと(?) :p) the differences are actually due to the slight variations in meaning, but I don't really have anything to back me up on that.
Note in the interest of transparency: Much of my answer (including the examples used) is based the explanation in this textbook on a Chinese website. As the site in Chinese, that's about as much as I can tell you about it. :)