The integer bounds you gave don't match the ASCII codes. For example, 'H' is 72.

As the commenters suggest, instead of reading up on the ASCII table, you should use char literals. So,

if ( ( inputString[i] >= 'A' && inputString[i] <= 'Z' )
        || ( inputString[i] >= 'a' && inputString[i] <= 'z' )
        || ( inputString[i] >= '0' && inputString[i] <= '9' ) ) {

You could also avoid all of this by using isalnum from ctype.h.

Write a function that takes a char and returns true if you want to remove that character or false if you want to keep it:

bool my_predicate(char c);

Then use the std::remove_if algorithm to remove the unwanted characters from the string:

std::string s = "my data";
s.erase(std::remove_if(s.begin(), s.end(), my_predicate), s.end());

Depending on your requirements, you may be able to use one of the Standard Library predicates, like std::isalnum, instead of writing your own predicate (you said you needed to match alphanumeric characters and spaces, so perhaps this doesn't exactly fit what you need).

If you want to use the Standard Library std::isalnum function, you will need a cast to disambiguate between the std::isalnum function in the C Standard Library header <cctype> (which is the one you want to use) and the std::isalnum in the C++ Standard Library header <locale> (which is not the one you want to use, unless you want to perform locale-specific string processing):

s.erase(std::remove_if(s.begin(), s.end(), (int(*)(int))std::isalnum), s.end());

This works equally well with any of the sequence containers (including std::string, std::vector and std::deque). This idiom is commonly referred to as the "erase/remove" idiom. The std::remove_if algorithm will also work with ordinary arrays. The std::remove_if makes only a single pass over the sequence, so it has linear time complexity.