If a non Jewish woman converts in order to marry a Jewish man are their children still considered Jewish? Even though the mother is converted and not raised in the religion.
If a non-Jewish woman converts to Judaism, she is considered to be naturalized1. This is of course assuming she had good intentions in her conversion. If the person in question never had intentions of keeping mitzvot (commandments) after becoming Jewish there can be complications.
But assuming the woman underwent an Orthodox conversion underneath a valid beit din (Jewish court) and was committed to living the rest of her life as a Jew, if she married a Jewish man and had children their children would be one hundred percent Jewish, no doubt about it.
Note that if the Jewish man she wishes to marry is not Torah observant, and is not willing to become so, there could be complications in the woman's conversion process.
Also, obviously someone wishing to convert to Judaism should discuss their own personal situation with their host rabbi. That way, he will be able to address the situation better.
1. I.e. as if she were born Jewish.
Echoing Ezra's excellent answer, if a candidate is deemed to be sincere despite marriage being a significant motivating factor in her conversion, she is considered wholly Jewish just as if she had been born Jewish once the conversion is complete, except that she cannot marry a Kohain, who are a small subset of Jewish men.
From the Rabbinical Council of America (the main body setting standards and practices for Orthodox Judaism in the United States to the extent that there is one) website (italics my own):
a. Where the Conversion is Primarily for the sake of Marriage
i. Where marriage to a particular Jewish partner is a major incentive to a prospective conversion, there is an increased possibility that the geirus may come with less than the complete commitment necessary for a conversion that would be in keeping with the standards we are trying to set for the regional Batei Din. Nonetheless, experience also shows that such a motivation can result in converts of the highest caliber. Conversion for the sake of marriage therefore requires the Beit Din to constantly reevaluate if the candidate and future partner are likely to subscribe to the requisite beliefs and practices. The Beit Din must be convinced that if the potential spouse were to disappear from the candidate’s life, his or her commitment to the Jewish faith and people would not waver. These factors inevitably prolong the process and make examination of the prospective convert more intense. Indeed, should the couple mention a proposed wedding date as a deadline or goal, the Beit Din should respond that the process will take significantly longer than that.