I am not sure your dummy coefficients have any clean interpretation at all any more, although of course they are OK to be used for predictive purposes. There's a formal interpretation of he coefficient showing whatever's left after accounting for all other variables, but identification does make things complicated, as you don't have a well-defined reference category anymore.
If you know the functional form of the trend (linear, quadratic, changepoint), you'd be better off adding that trend directly to your explanatory variables.
Results from an ordered logit/probit regression are always unintuitive, but categorical explanatory variables are as meaningful as continuous ones. I'd even say that they are easier to interpret.
For a concrete example, you could look at Dobson, An Introduction to Generalizer Linear Models, 2002, 2nd ed., Chapter 8. In her "car preferences" example, the dependent variable is the importance of air conditioning and power steering (three levels: "no or little importance", "important", "very important") and the two explanatory variables are gender (male or female, coded as 1 and 0) and age (18-23, 24-40, >40, coded as age2440 = 1 or 0, and agegt40 = 1 or 0).
Fitting an ordered probit model you get (I've used R, MASS library, polr() function):
Coefficients:
male age2440 agegt40
-0.3467 0.6817 1.3288
Intercepts:
NoImp|Imp Imp|VeryImp
0.01844 0.97594
Then you can compute the probabilities for women (male = 0) over 40 (age2440 = 0, agegt40 = 1):
NoImp Imp VeryImp
0.095 0.267 0.638
and for men over 40 (male = 1):
NoImp Imp VeryImp
0.168 0.330 0.502
Their difference is the gender partial effect:
NoImp Imp VeryImp
-0.073 -0.063 0.136
I think that it's meaningful ;-)
Best Answer
Generally your are estimating probabilities for every category j of your dependent variable y. Similar to marginal effects, not as far as I know. You can estimate the probabilites for the response-categories with mfx in stata if I remember correctly.
Concerning the interpretation of the coefficients UCLA can help: "Standard interpretation of the ordered logit coefficient is that for a one unit increase in the predictor, the response variable level is expected to change by its respective regression coefficient in the ordered log-odds scale while the other variables in the model are held constant."