Achika's death is explained differently in the two continuities I am familiar with. Below are the two generally accepted theories that explain the death of Achika.
In the TV series, her early death is assumed to be a result of fighting Kain. In her battle with Kain, and her efforts to protect Tenchi, she expended much of her Jurai power, and as a result significantly shortened her lifespan. This seems to be just a theory, though. It is possible however, since we know a Jurai emperor died dispelling the same energy as Achika in their efforts to capture and contain Kain. It is because Jurai energy was used in his capture that Kain Despised it so much and came to destroy Achika. Perhaps in trying to destroy her, he weakened her enough to cause her early death...
The OAV differs since in it Kain never makes an appearance. (In fact, this theory seems to have been based on a Novel, though I am not sure.) It suggests that Achika may have died because of a broken heart, at least partly. As a young girl, she showed great promise, being able to communicate with Funaho, one of the Jurai trees, and see spirits. Yosho wanted her to inherit the sword (I'm assuming the Tenchiken) and the power of the Jurai royal family. Achika, though, fell in love with a young man and wanted to be a normal girl. She didn't want to disappoint Yosho. She tried her best to be what Yosho wanted, but Failed in a ritual with Funaho (Jurai royal tree, remember) and got seriously wounded. During her recovery, the man she loved betrayed her. She did recover from the wound, weak, and married a man older than herself because he looked like Yosho. This man was Nobiyuki. Achika seemed happy with the marriage. A year later Tenchi was born, and 4 years after his birth, Achika died. She said she'd be with Tsunami, and told Yosho to train Tenchi until he was 15.
Now, to be completely honest, I don't believe that the OAV version of her death is correct. Tenchi mentions in the English dub that Achika's Jurai power passed to him and that the Jurai power he inherited could have enabled Achika to live a longer life. This hints to me that when Achika died, she had no powers at all, so excessive drain would not be an issue. Further, we have seen that Jurai power can lengthen the lifespan of a half human (Yosyo). In the Tenchi: In Love movie, Achika is in very close proximity to the blast from the Dimensional Cannon, and I find it more likely that this causes her early death. I also think she could have died a normal mundane death that normal people without Jurai powers do. Don't you think you might live longer if you could put up a force-field when needed?