Are you referring to using an independent variable that is ordinal, with more than two values? If so, whether or not you use dummy variables depends. Are you willing to accept that the difference in effect between, say the first and the second values is the same as the difference in effect between, say the fourth and the fifth values? Its very unlikely that we ever except this, and so I would recommend dummy variables.

Also, depending on the question you want to answer, I'd say there are more methods than just logistic regression. Off the top of my head: Poisson, log-link binomial, and probit are a few. You should look into those too.