Omnyouji's (being able to protect people from evil spirits, summon Shikigami etc) are quite true in Japanese folklore. For example, unless I have my info wrong... Abe no Seimei, whom I mentioned before, did actually live, and he was apartenly one of the most powerful Omnyou of his time. Again, unless my have my info wrong, the Omnyou way was quite popular in Heian period Japan (around the year 1000) and were often used to protect the royal court and the emperor.
Omnyou are often compared to the Western concept of a Wizard, and their Shikigami often compared to a Wizard's familar.
The way of the Omnyou are often said to be linked to Taoism and Buddism, more than Shinto.
So far, the best anime/manga I've seen dealing with the Omyou way is without a dobut is Tokyo Babylon. All three of the main characters practice the art in some form or another, and the majority of the story revolves around it that fact.
Some other anime with Omnyou references in it would be Abenobashi (of course, it's all very much so warped for comedic effect) X (of course, being a sequel to Tokyo Babylon) Blue Seed and even Spirited Away. (the paper bird that follow Chihiro could very well be Shikigami)
A fairy brief explanation of Omnyou, Shikigami and their references in anime and manga can be found here: