Do males of all sexual species have Y chromosomes?
Very short answer
No, not all males of all sexual species have Y chromosomes. You might want to have a look to the Wikipedia page on sex-determination systems.
Long answer
Diversity among the species that reproduce sexually
Not all species that have sexual reproduction have sexes. Yeasts, for example, have mating types but no sex.
Diversity among the species that have sexes
Sex is determined by both genetic and environmental factors. In some species, genetic factors are more important than environmental factors in other species it is the reverse. Species which sex is mostly determined by the genetics are said to have GSD (Genetic Sex Determination). For example, humans are GSD, as the female is XX and the male is XY. The species where sex is mostly determined by the environment are called ESD (Environmental Sex Determination). For examples, crocodiles are ESD as sex is determined by temperature. It is important to understand however that there is a whole continuum between these two extremes.
Diversity among the species that are GSD
Among the species that are GSD, some have sexual chromosomes some don't. Some have one locus  (locus=position on a chromosome) that determines the sex, some have many loci (loci=plural of locus). Humans, for example, have sexual chromosomes (X and Y) and have only one locus which determines the sex. This locus is called SRY and it codes for a protein called TDF.
Now you can split GSD with sexual chromosomes into two more categories (it is a bit more complex in reality): XY and ZW. XY are those species where the male is heterogametic (XY), while the female is homogametic (XX). In ZW systems, the male is homogametic (ZZ) and the female is heterogametic (ZW). Birds and some plants have ZW systems for example, while mammals (except &quot;basal&quot; mammals) and Drosophila have XY system.
See also the post What determines sex in birds?
Extra Information
Dosage Compensation
In species that have sex chromosomes, there is a difference in the number of copy of genes between the sexes. In eutherian mammals, for example, females have two copies of all the genes on the X chromosomes, while males only have one copy of most of these genes (plus a few Y chromosome genes). The set of methods to deal with this issue is called Dosage Compensation and there is also an impressive diversity dosage compensations.
Comments on this diversity of sexual systems
The diversity in a sex-determination system, dosage compensation and other things related to sex are impressive. It is even more impressive when we look at how many independent origins there are. Below are some other examples.
The Amazon molly (a fish) is a species that have sexual reproduction but there are no males. The females have to seek for sperm in a sister species in order to activate the development of the eggs but the genes of the father from the sister species are not used. (see this article)
There are also hermaphrodites including sequential hermaphrodites (first male, then females or the opposite) in plants and animals. There are also species where populations are made of hermaphrodites and females and others where there are hermaphrodites and males (very uncommon).
In some species, the sex is determined by social factors. In clownfish, the sex is determined by comparing its own size with the size of the other fishers living in the same anemone.
In an ant species (or two species actually), males and females can both reproduce by parthenogenesis (some kind of cloning but with meiosis and cross-over) but they the meet they reproduce together and their offsprings are sterile workers. So males and females are just like two sister species that reproduce sexually to create an army to protect and feed them. See more information in this paper

Here is a nice figure from Bachtrog et al. 2014 that offers an idea of the diversity of sex determination system (thank to @rg225 for pointing out this figure).

Book suggestion
The Evolution of Sex-Determination is a great book that may interest you.